Inclusion Criteria:

* A prospective patient for allogeneic BMT for malignant hematologic conditions. A patient with history of CNS involvement is eligible if CNS disease is in remission at time of study consideration.
* The donor and recipient must have an HLA-match (8/8 HLA-A, -B, -C, and -DRB1) or haploidentical match (per protocol criteria). High resolution typing is required for all alleles.
* Diagnoses to be included:

1. Acute Leukemia in remission. Remission is defined as the absence of blasts in the peripheral circulation at the time of enrollment, \< 5% blasts in the bone marrow and absence of extramedullary disease including CNS involvement.
2. Chronic Myeologenous Leukemia (CML) in first or subsequent chronic phase failing to respond (or intolerant) to at least two different tyrosine kinase inhibitors. CML in accelerated or blast phase (CML-AP/BP) are eligible without requirement to fail tyrosine kinase inhibitor therapy, but must be in remission at time of enrollment. Remission is defined as the absence of blasts in the peripheral circulation at the time of enrollment, \<5% blasts in the bone marrow and absence of extramedullary disease including CNS involvement.
3. Myelodysplastic syndrome (MDS) with intermediate or high-risk IPSS or equivalent IPSSR score with \< 10% blasts in the bone marrow.
4. Mature B Cell Malignancies (including Mantle Cell Lymphoma, Follicular Lymphoma. Diffuse Large B Cell Lymphoma, Non-Hodgkin Lymphoma not otherwise specified).Subjects should have extinguished standard of care options prior to being considered eligible for this trial
* Subjects aged 3 to 39 years
* Lansky/Karnofsky Performance Scale score of 70% or higher
* Life expectancy of greater than 6 months
* Subjects must have normal organ and marrow function (as defined in protocol)
* Ability to take oral medication and be willing to adhere to the vorinostat regimen
* For females of reproductive potential and men: The effects of vorinostat on the developing human fetus are unknown. For this reason and because histone deacetylase inhibitor agents as well as other therapeutic agents used in this trial (e.g., calcineurin inhibitor \[tacrolimus or cyclosporine\], methotrexate, mycophenolate, and cyclophosphamide) are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Women must continue using contraceptives for at least 6 months after the end of therapy and men must continue contraceptive use for 3 months after completion of vorinostat administration.
* Ability to understand and the willingness to sign a written informed consent document
* Stated willingness to comply with all study procedures and availability for the duration of the Study
* For the cognitive assessment and patient-reported QOL exploratory correlative portion of the study, subjects and caregiver must speak, read and understand English. Subjects who are too young to read must be able to understand and speak English, age-appropriately. Subjects who do not speak, read and understand English but satisfy all other inclusion criteria may still participate in the study but will not complete the cognitive and QOL portions.

Exclusion Criteria:

* Subjects who are not a candidate for an allogeneic BMT based on the current local site institutional BMT program clinical practice guidelines. Organ function criteria will be utilized per the current local site institutional BMT program clinical practice guidelines. There will be no restriction to study entry based on hematological parameters.
* Presence of anti-donor HLA antibodies (per protocol criteria).
* Subjects who are enrolled on another GVHD treatment or prevention trial.
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to vorinostat.
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. Subjects still under therapy for presumed or proven infection are eligible provided there is clear evidence (radiographic findings and/or culture results) that the infection is well-controlled. Subjects under treatment for infection will be enrolled only after clearance from the PI.
* Pregnant women are excluded from this study because vorinostat is a histone deacetylase inhibitor agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with vorinostat, breastfeeding should be discontinued if the mother is treated with vorinostat. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately.
* Subjects with evidence of HIV seropositivity and/or positive PCR assay, HTLV1/HTLV2 seropositivity. The safety of allogeneic HSCT is not yet well-established for this population.
* Subjects with evidence of Hepatitis B or Hepatitis C PCR positivity. Hepatitis reactivation following myelosuppressive therapy can lead to fatal complications.
* Subjects with a history of prolonged QTc syndrome.
* Subjects who have had prior treatment with a drug like vorinostat (i.e., valproic acid) within the last 30 days.
* Subjects with documented evidence of cognitive impairment prior to enrollment on this study (diagnosis of dementia, mild cognitive impairment, or other neurological illnesses that impacts cognition) are excluded from the cognitive assessment portion of the study only.