Study to Assess OPB-31121 in Advanced Leukemias or Myelodysplastic Syndromes

The Study Drug: In studies that have been performed in animals, OPB-31121 has been shown to stop certain changes to proteins in the body. Stopping these changes may have an effect on tumor cells. How OPB-31121 works is not fully known. Study Groups: The first 3 participants enrolled in the study will receive the lowest level of OPB-31121. Each additional group of 3 or 4 participants will receive a dose level dependent upon how the previous group responded to the study drug. You may have up to 13 study "cycles" (about 273 days). During each 28 day study cycle, you will receive the study drug for 21 days and then "rest" (no study drug) for 7 days. The cycle is 28 days, 21 days of drug, 7 days of rest. Study Drug Administration: The study drug is a pill that should be taken at the same time each day. You should drink 8 ounces (1 cup) of room temperature, non-carbonated water when you take the study drug. You will eat a snack or meal within 30 minutes after taking the study drug. You should remain sitting upright or standing for at least 30 minutes after taking the drug. Study Visits for AML, ALL, or MDS participants: On Day 1 of Cycle 1, you will have the following tests and procedures performed: * You will have a physical exam, including measurement of your vital signs. * You will have an ECG. * Urine will be collected for routine tests. * Women who are able to have children will have a urine pregnancy test. * Blood (about 6 tablespoons total) will be drawn about 10 times. These blood draws will be for routine tests, heart, thyroid testing, pharmacokinetic (PK) testing, and genotyping. PK testing measures the amount of study drug in the body at different time points. Genotyping tests are tests that look at pieces of your DNA called genes. These tests will look at some specific genes that may predict how your body may break down certain types of drugs. A small needle called a "heparin lock" may be inserted into a vein to make these multiple blood draws easier. On Day 2 of Cycle 1, blood (about 2 tablespoons) will be drawn for heart, PK, and biomarker testing. On Days 8, 15 and 21 of Cycle 1: * You will have a physical exam, including measurement of your vital signs. * You will have an ECG. * Urine will be collected for routine tests. * Blood (about 2 tablespoons) will be drawn for routine, heart, and thyroid testing. * Blood (about 6 tablespoons total) will also be drawn for PK testing. Blood will be drawn a total of 10 times, with the last blood draw about 10 hours after you receive the study drug. * Blood (about 1/2 tablespoon) will be drawn for biomarker testing (Day 21 only). On Day 22 of Cycle 1, blood (about 1 tablespoon each time) will be drawn 2 times for PK testing. On Day 23 of Cycle 1, blood (about 1 tablespoon) will be drawn 1 time for PK testing. On Day 28 of Cycle 1, you will have the following procedures performed: * You will have an ECG. * You will have a physical exam, including measurement of your vital signs. * You will have a bone marrow aspiration sample taken to check the status of the disease. This sample will be taken after you have taken the study drug and only at the end of each odd cycle. On Day 1 of Cycles 2-13, you will have the following tests and procedures will be performed: * Blood (about 1-1/2 tablespoons) and urine will be collected for routine, heart, and thyroid tests. * You will have a physical exam, including measurement of your vital signs. * You will have an ECG. * Women who are able to have children will have a urine pregnancy test. * On Day 1 of Cycles 2 and 3 only, blood (less than 1 tablespoon) will be drawn for biomarker tests. * You will have a MUGA scan. * If your doctor thinks it is needed, you will have a bone marrow aspiration/biopsy. Study Visits for CLL or CML participants: On Day 1 of Cycle 1, you will have the following tests and procedures performed: * You will have a physical exam, including measurement of your vital signs. * You will have an ECG. * Urine will be collected for routine tests. * Blood (about 8 tablespoons total) will also be drawn a total of 10 times for routine, heart, thyroid, biomarker, genotyping, and PK testing. The last blood draw will be 24 hours after you receive the study drug (Day 2). A small needle called a "heparin lock" may be inserted into a vein to make these multiple blood draws easier. * Women who are able to have children will have a urine pregnancy test. * If your doctor thinks it is needed, you will have a bone marrow biopsy. On Day 2 of Cycle 1, blood (about 2 tablespoons) will be drawn 24 hours after your study drug dose for heart and PK testing. On Days 8 and 15 of Cycle 1, blood (about 7 tablespoons) will be drawn for routine and PK testing. On Days 21 of Cycle 1, you will have an ECG. Blood (about 8 tablespoons in total) will be drawn about 8 times for PK, routine, heart, thyroid, and biomarker testing. On Days 22 of Cycle 1, blood (about 1 tablespoon each time) will be drawn 2 times for PK testing. On Day 23 of Cycle 1, blood (about 1 tablespoon) will be drawn 1 time for PK testing. During Cycles 2-13, blood (about 1 tablespoon) will be drawn every 2 weeks for routine, heart, and thyroid testing. On Day 1 of Cycles 2-13, you will have the following tests and procedures performed: * You will have a physical exam, including measurement of your vital signs. * You will have an ECG and MUGA scan. * Blood (about 2 tablespoons) will be drawn for routine, heart, thyroid, and biomarker tests. * Women who are able to have children will have a urine pregnancy test. * During Cycles 2-3 only, blood (about 1 tablespoon) will be drawn for biomarker testing. End-of-Study Visit: Once you are off study, you will have an end-of-study visit. At this visit, the following tests and procedures will be performed: * You will have a physical exam, including measurement of your vital signs. * You will have an ECG. * Blood (about 7 tablespoons) will be drawn for routine, heart, thyroid, and PK tests. * Urine will be collected for routine tests. * If you have AML, ALL or MDS, blood (about 1-1/2 tablespoons) will be drawn for biomarker testing. Follow Up: Your doctor or other study staff will call you about 30 days after your final visit to see how you have been feeling. The phone call will take about 5-10 minutes. This is an investigational study. OPB-31121 is not FDA approved or commercially available. At this time, OPB-31121 is only being used in research. Up to 48 patients will take part in this multi-center study.