I've applied for and received a visa at the Mongolian Embassy in Bangkok and found the whole process to be straightforward.
The following applied (may 2014):
I'm a Dutch citizen. Visa application before 11AM, pick-up 3 work-days later after 4PM.
Costs 1705 Bath per visa/person.
The application form can be requested via email or received at the application desk. You will need one photo ID and will have to bring your passport (which will be kept at the embassy for the duration of the application). Payment had to be made at a SCB Bank, as a deposit. I had to bring proof (bank-slip) of the deposit to retrieve my passport with visa. Everything was well explained in English at the embassy.
Double-check the embassy address. As they had moved prior to my application, I had some trouble finding the right address. At that time I noted down the following address details (possibly outdated):
Embassy of Mongolia
100/3 Soi Ekkamai 22, Sukhumvit 63,
Klongton Nua, Wattana, Bangkok 10110
Phone 02-381-1400, 392-1011
Opening hours: monday-friday between 9.00-12.00 and 14.00-16.00
Good luck and enjoy your trip!