If you are talking about UK visa, then the answer is that you should be fine, although it could make travel easier if you update/change details on your visa. I am speaking from experience.
When my wife and I got married, we both lived in the UK; I had a USA passport with an HSMP visa (then-equivalent of Tier 1 General) and my wife had a Russian passport with a Work Permit visa (then-equivalent of Tier 2). After we got married, she changed her name and got a new Russian passport showing her new name. After that, whenever she had to travel internationally, she would carry her old (cancelled) Russian passport with the visa, her new Russian passport and a copy of the marriage certificate (we got married in the USA). Not once has she had any problems entering the UK. This continued for about a year until she got her spouse visa in her new name in her new passport.
Depending on visa type, you may have had to register with a local police department. If this is the case, you must notify the local police department of the change of name (and this was the case with my wife, too) - but you still are not required to get a new visa.
It is important to carry with you both, old and new, passports and some sort of an official document confirming the change of your name (marriage certificate, court ruling, deed poll, etc.)