Can I enter Romania from Dubai with a Schengen visa?

Yes you can! This is how to do:

1. You have to have a valid Schengen Visa
2. You are eligible to stay up to 5 days in Romania
3. You have to show the immigration that your next destination is a Schengen Country (ex: Greece, France, etc.) and in this case your trip should be Dubai - Romania - Athens (for transit) - Dubai

This type of visa is called a transit Visa which is valid only for 5 days in Romania as long as you travel to a Schengen country afterwards.

For the details of the policy, see the Romania Ministry of Foreign Affairs website.

From Wikipedia:

A Schengen visa is only valid for the Schengen Area. For individuals who require a visa for Bulgaria, Croatia, Cyprus and/or Romania, a separate Bulgarian/Croatian/Cypriot/Romanian visa has to be obtained. Note that holders of a Schengen visa can enter Bulgaria and Croatia for up to 90 days in a 180 day period and/or transit through Romania for up to 5 days during the validity of their Schengen visa without having to apply for Bulgarian, Croatian and/or Romanian visas.

If you have a Schengen Visa with 2 or multiple entries then you can get into Romania even if you don't want to go a Schengen State afterwards. This kind of stay is limited to only 90 days.

Here is the source : http://www.mae.ro/en/node/2040

In case you hold passport of any of the countries listed in here (http://www.mae.ro/sites/default/files/file/pdf/formulare-consulare/Vize/2014.06.09_annex_2_en_conf_reg_509-2014.pdf) then you don't need any kind of visa to travel to Romania. ANSWER

I know this answer already exists, but I wanted to add sources for reference and because I cannot comment on previous answers because of reputation so I added another answer