As previously mentioned, Russian doesn't use articles (a, the), so Russian speakers use them - or don't - by guesswork, and often get them wrong.
What I haven't seen anyone else mention, however, is that the present tense of to be (I am, thou art†, he is, we are, you are, they are) is rarely (if ever) used in Russian. As a result, again, Russian speakers sometimes make surprising mistakes in this area. (My favorite: "Is there is...?")
In speech, of course, there are at least three major pitfalls: Russian lacks a "th" sound - foreign words that are imported into Russian tend to get substituted with "f" or "t". When speaking English, "th" tends to turn into "s" or "z". If you're feeling especially cruel, ask your Russian colleague to say "thither". (Of course, a lot of Americans also have trouble with that one.)
Russian also doesn't have an equivalent to English "h" - the Russian letter х, pronounced like the "ch" in loch, is not equivalent - so foreign (mostly German) words imported into Russian usually substitute "g". Russians speaking English will, at first, turn all of their aitches into gees; later on, some learn to pronounce an English h, while others convert h's into х's - the source of the infamous "kheavy Roossian excent".
Finally, several of the "short" English vowel sounds - the a in "at", i in "in", and u in "up" - don't exist in Russian, while Russian has at least one vowel sound (ы) that doesn't exist in English. (Hence "excent" instead of "accent".)
†Yes, I know - we don't actually use "thou" anymore. Russians do, however (ты) and so I mentioned it for completeness.