When someone is perceived to speak English with a foreign accent, there are at least two different elements involved. The first is what the American Heritage Dictionary explains as
A characteristic pronunciation, especially:
1. One determined by the regional or social background ofthe speaker.
2. One determined by the phonetic habits of the speaker'snative language carried over to his or her use of another language.
This is what most accent reduction programs seem to focus on: individual vowel and consonant sounds, syllabic stress, etc. Another element, however, is the set of characteristic grammatical errors made by speakers whose native language is not English. It is this aspect that allows me to detect when something I'm reading has been written by a native Chinese speaker, for instance. Because Chinese has no articles and no inflected forms for plural nouns or verb tenses, native Chinese speakers frequently omit these constructions when speaking or writing English. "Yesterday I went to the store and bought an apple for fifty cents" would become, "Yesterday I go to store and buy apple for fifty cent".
Examining these characteristic errors can also provide insight into the grammar of a particular language. I've never studied Russian, but I've noticed that many Russian speakers make excessive use of the present progressive. My violin teacher, who grew up in Moscow, says things like, "When I am practicing this passage, I am thinking about relaxing this finger" whereas a native English speaker would use the simple present, i.e. "When I practice this passage, I think about relaxing this finger". I finally began to wonder whether there's something about the Russian language that emphasizes the progressive aspect of verbs, and sure enough, there is.
In elementary school grammar we usually learn only about a verb's tense, which is simply where the action occurs in time, i.e. past, present, or future. But verbs have another quality called aspect, which describes the state of completion of the action. It can be incomplete (imperfect), as in "I was going to the store, or complete (perfect), as in "I went to the store". This is the difference between the Spanish imperfect ("iba a la tienda") and preterite ("fui a la tienda"), or the imperfect and aorist in Ancient Greek. Based on the superficial knowledge of Russian verbs I've managed to gain by perusing sites like this one and this one, in Russian the aspect of a verb is more important than its tense, and most verbs come in imperfective-perfective pairs. I'm guessing that when Russian speakers learn English, they're somehow taught that the equivalent for their imperfective verb is formed by the English present progressive (to be + present participle), whereas the simple present tense is much more idiomatic and often conveys that progressive aspect implicitly.
By the way, the term "perfect" when used to describe a verb has nothing to do with the connotation of "flawless", as I wondered when I first encountered the term many years ago. They do have related etymology, however, from the Latin "perficio" meaning "to complete". The word seems to have evolved to its more common meaning in that something completed, or brought to full development, is presumed to be of high quality.