If you are from the below countries/ regions- the official website of Immigration, Bangladesh Police shows that you are eligible for Visa Exemption for upto 90 days. 

Visa Exempted Country List for Bangladesh Visit

Barbados – Visa exempted for upto 90 days

Bhutan – Visa exempted for upto 90 days

China (Service or Official Passport or Public Affairs) – Visa exempted for upto 90 days

Fiji – Visa exempted for upto 90 days

Gibraltar – Visa exempted for upto 90 days

Grenada – Visa exempted for upto 90 days

Ireland – Visa exempted for upto 90 days

Jamaica – Visa exempted for upto 90 days

Saint Kitts and Nevis – Visa exempted for upto 90 days

Trinidad and Tobago – Visa exempted for upto 90 days

Source : Immigration, Bangladesh Police 

http://www.immi.gov.bd/