Two things that I notice, at least in caricatures of non-Japanese speakers speaking Japanese, are mistakes in the vowel sounds and in the cadence of words.
The Japanese vowel sounds do roughly correspond to some of the sounds in English, but they take a while to get right. As an extreme example, consider the pronunciation of the American English word "karaoke". The first part sounds like "carry" and the second like the first half of "okie dokie". In comparison, the pronunciation of the Japanese word "karaoke" is closer to someone saying the words "car are OK" (although that's still not perfect, and it may depend a lot on your accent).
For an example of a (Japanese person voice acting as a) Canadian character speaking heavily accented Japanese, this anime clip shows a lot of what I'm talking about, especially in her "Sew narn dez kar?" lines (for those following along at home, that's meant to be "Sou nan desu ka?")
This is a little harder to explain, especially if you haven't learned much Japanese yet, but it relates to the discussion about stressing the right parts. Japanese is a language where words are made up of very specific syllables, which means that words have a particular rhythm to them based on those syllables, and it's very easy to pronounce words with a more English rhythm until you get used to it.
This is not unusual - there are words in English where their more common modern pronunciation is different to what would have traditionally been considered "correct". A great example of this is "Wednesday", which most people would say as "wenns dei" (two syllables). However, if you look for some old BBC announcers saying it (since they used to have very strict rules about pronouncing words like this, so they're a good example for these things), you'll find that there are three distinct syllables - "wed nns dei".
In Japanese, each hiragana character you use to write a word is effectively one "beat" of the word when you say it, and although the specific timing can vary a bit (especially in casual speech), there are some very distinctive non-native things that crop up.
For example, Kyoto, or to romanise it more accurately, kyouto, which consists of 3 "beats" - kyo-u-to (although the -u- actually has the effect of lengthening the o sound before it, so you wouldn't emphasise an "oo" sound when pronouncing it). A common, but bad, pronunciation, separates the first sound into two and makes it more like "kai-yo-to", almost like the start of "coyote"; slightly better but still wrong is "kee-yo-to". The "k" and "y" sound are part of the same beat, and should not be separated.
One set of words I have trouble distinguishing, part of which comes down to not being able to hear the distinct timing of, are the words "kon'yaku" (engagement, as in engagement ring), "ko'nyakku" (cognac) and "kon'nyaku" (konjac jelly). In "kon'yaku", the sounds are "ko-n-ya-ku", noting in particular that "n" gets a beat to itself. In "ko'nyakku", the sounds are "ko-nya-k-ku" - the double consonant is a glottal stop, which gets a beat to itself, while the "nya" sound is a single beat. And in "kon'nyaku", the sounds are "ko-n-nya-ku", so there's a beat for "n" and a beat for "nya". However, if you read those words without the deliberate punctuation to help you see where to break, then a naive reading of "konyaku" versus "konyakku" may end up with you splitting them along the same lines and getting at least one of them wrong (either "ko nyaku" or "kon yakku").