Inclusion Criteria:

* The following diagnoses will be eligible for this study:
* Myelodysplastic syndromes: the diagnosis of MDS must be confirmed by a bone marrow aspirate and/or biopsy within two weeks prior to registration; NOTE: blast count must be < 20%; patients with any International Prognostic Score (IPSS) are eligible; patients with low or intermediate (INT)-1 IPSS must have a platelet count < 50,000/mm\^3 and/or absolute neutrophil count (ANC) < 500/mm\^3 within seven days prior to registration
* Chronic myelomonocytic leukemia (dysplastic subtype): the diagnosis of CMMoL must be confirmed by a bone marrow aspirate and/or biopsy within two weeks prior to registration; patients with CMMoL must have a WBC < 12,000/mm\^3, documented within 4 weeks prior to study entry (two sets of counts that are 2 weeks apart will be taken)
* Acute myeloid leukemia with multilineage dysplasia: the diagnosis of AML-TLD must be confirmed by a bone marrow aspirate and/or biopsy within two weeks prior to registration; NOTE: there must be evidence of >= 20% blasts on the review of the bone marrow aspirate and/or biopsy; AML-TLD will be interpreted to include patients formerly diagnosed by French-American-British (FAB) criteria as refractory anemia with excess blasts in transformation (RAEB-t), as well as patients with no history of antecedent hematologic disorder who have AML which meets criteria for AML-TLD by World Health Organization (WHO) criteria; patients with AML-TLD must have a white blood cell (WBC) =< 30,000/mm\^3 documented within 4 weeks prior to study entry (two sets of counts that are 2 weeks apart will be taken); patients whose WBC has doubled within this period of time and is greater than 20,000/mm\^3 at the time of screening will not be eligible
* Women must not be pregnant or breast-feeding; all females of childbearing potential must have a blood test or urine study within two weeks prior to registration to rule out pregnancy
* Women of childbearing potential and sexually active males must be strongly advised to use an accepted and effective method of contraception
* Patient must have Eastern Cooperative Oncology Group (ECOG) performance status between 0-2
* Patient must have no prior treatment with azacitidine, decitabine or entinostat
* Patients must not have active infections at the time of registration
* Serum creatinine < 2.0 mg/dL; test must be done within seven days prior to registration
* Total serum bilirubin within institutional limits unless due to intra- or extramedullary hemolysis or Gilbert's syndrome; test must be done within seven days prior to registration
* Aspartate transaminase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine transaminase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =< 2.5 x institutional upper limit of normal (ULN); tests must be done within seven days prior to registration
* Patients must not have received any AML induction chemotherapy or stem cell transplantation; any other treatment for their disease, including hematopoietic growth factors may not be given, within three weeks prior to registration, and should have recovered from all toxicities of prior therapy (to grade 0 or 1)
* Patients must have no clinical evidence of central nervous system (CNS) or pulmonary leukostasis, disseminated intravascular coagulation, or CNS leukemia
* Patients must have no serious or uncontrolled medical conditions
* Patients who have therapy-induced MDS, CMMoL (dysplastic) and AML-TLD are eligible and will be treated as separate cohorts from the patients with de novo MDS, CMMoL (dysplastic) and AML-TLD
* Patients should have a life expectancy of at least six months
* Patients must not have advanced malignant hepatic tumors
* Patients must not have a known hypersensitivity to azacitidine or mannitol
* Southwest Oncology Group (SWOG) ONLY: all SWOG patients must be registered on SWOG-9007 ("Cytogenetic Studies in Leukemia Patients"); collection of the pretreatment bone marrow specimen (or of peripheral blood if the marrow is not aspirable) must be completed within 28 days before registration; the pretreatment specimen must be submitted to a SWOG-approved cytogenetics laboratory as described in protocol SWOG-9007; note that submission of bone marrow cytogenetic studies are required to calculate the IPSS score (stratification issue); in addition, cytogenetic response will be measured at follow-up requiring a second cytogenetic study at the end of protocol treatment; NOTE: In addition to SWOG-9007, SWOG patients must be offered participation in S9910, the leukemia centralized reference laboratories and tissue repositories ancillary study; If consent is given, collection of pretreatment blood and/or marrow specimens must be completed within 14 days prior to registration. If the patient consents to participate in S9910, pretreatment specimens of marrow and/or peripheral blood must be submitted to the Southwest Oncology Group Myeloid Repository at the University of New Mexico for cellular and molecular studies; S9910 also requests submission of remission and relapse specimens