As an Austrian, I can confirm that Arnold Schwarzenegger sounds funny no matter whether he speaks English or German (or rather tries to). When speaking German, he has more of an American accent than most other Americans I know. The reason may be that he is from Styria, a certain part of Austria where all people sound kind of funny to other Austrians, so why shouldn’t that apply to any other language they learn.
In general, to elaborate on what some of you have already posted, there is a certain (very large) set of sounds which humans can produce. Each of the languages uses only a very small subset of these sounds. A child has no preference for a certain subset of the sounds. Learning a mother-tongue (or more than one) basically means to loose the ability to recognize the sounds not used in the language(s), in order to focus on the sounds that are used and distinguish them better. A lLanguage learned during childhood usually become a mother-tongue, a language learned later in live usually becomes a foreign language. When speaking a foreign language, we mimic the sounds that make up this language using the sounds we know from our mother-tongue(s). We have no chance doing better, because we can hardly hear the other sounds, let alone reproduce.
Of course there are some people around who are very gifted or do not loose their ability to acquire yet another mother-tongue, but they are the exception.
As an example, there are really many ways how the letter ‘o’ can be pronounced. In the German language (my mother-tongue), only three or four of these sounds are actually used. Only recently did I manage to HEAR the ways this letter is pronounced in English (though maybe not all the ways in all areas where English is spoken), some of which sound to me like a mixture of ‘o’ and ‘a’. But when I try to reproduce that, a speaker of English usually thinks I just say ‘a’ and may be confused. So I stick to my German ‘o’ and have an accent, but at least I am understood.
Another thing is the intonation of speech, which varies considerably between languages, but is rarely ever addressed in language courses. We have certain ways of intonating a sentence as statement, question, command and so on. However, when we apply these intonations, which we have learned for our mother-tongue, to other languages, this may sound funny at best, if not confusing. It is even harder to get rid of that kind of accent than it is with that cannot-reproduce-sound-exactly thing.