There are four situations to deal with when languages collide. I will use English and Russian as examples:

• Case one: A Russian speaking Russian. The native always speaks his own language correctly. Do so on your audiobook. Russian speakers will know. Find someone to give you a quick lesson and practice it until you say it right.
• Case two: A English-speaker speaking Russian. He talks with an accent, reflecting his native language and how well he learned to pronounce Russian. On your audiobook, be internally consistent. If Joe spoke Russian well yesterday, he must speak it well today too.
• Case three: A Russian speaking English. Here's where you can give a “Russian accent” to the character. To get a sense of that accent, go on the internet and search for audio feeds in English emanating from other countries. Listen to them report the news in English. If it is a native Russian speaking English, you will pick up subtle pronunciations and speech mannerisms. Download or tape some of them and practice sounding like them. If the English is very good, though, they may have hired a native English-speaker to read the news.
• Case four: Dialects of native English speakers. In this category are national and regional differences, and also social class distinctions. In the US, there are major differences between Northern and Southern dialects, and those can be subdivided (totalling about 24 geographic areas, not counting Alaska or Hawaii).

An internet search for 'dialect' will get you lots of hits, with sound-samples and You can find books to work with on certain dialects (to sound Irish, Australian, etc.). They provide a set of pronunciation rules, which will help you create the effect, if you follow them consistently.

If your story is set in another English-speaking country or another region of your own country, you may be considering trying to make your narrator 'sound like he is from there', especially if the story is told in the first person. Resist the temptation. Unless you can speak the dialect well enough to fool a native, you will constantly struggle to sustain a dialect consistently. A voice that goes in and out of dialect is very noticeable. Just sustaining one character's dialogue as “British English” or “Australian English” will take a lot of work on your part to get the sound right all the time.

Poorly done accents are immediately jarring to the listener. The listener's attention is diverted from what you said, to how you said it. Listeners who are from that region (or who are part of that ethnic group) may be offended if you stereotype their speech patterns due to only having a limited handle on how to sound authentic.

I have two recommendations:

• Don't use accents for your narrator's voice, regardless of setting. Use your normal speaking voice.
• Minimize your use of accents. You can convey just as much of an accent or dialect saying one or two words “that way” as you as you can the entire sentence. For example, pretend that you are an English butler, and you walk into the study where the lady of the house is sitting. You say: “Would you care for a cup of tea, madam?” Learn how to say one or two words with a British accent (like “care” or “tea”). If you do it well, it will carry the whole sentence.