Most English speakers use a different emphasis for certain words in a sentence. Speakers stress content and focus words, emphasizing them in order to increase meaning. An English sentence contains three types of words: a focus word, which is the most important word in the sentence that the rest of the sentence revolves around; content words, which add meaning to the focus word; and function words, which complete sentences grammatically.
Content words usually consist of nouns, verbs, adjectives, adverbs and numbers, and should be emphasized in speaking. The focus word tends to come last in a sentence and should be emphasized as well. Function words are those remaining words in a sentence and shouldn't be emphasized when speaking, as they're important to the grammar of the sentence but not the meaning.
Find some sentences that contain content, focus and functional words, and practice placing the emphasis on the proper words. A few example sentences are "Mary loves her new car" and "She likes to drive it around town." In the first sentence, the content words you'll want to emphasize are Mary, loves and car; in the second emphasis should be placed on drive, it and town.
Once you get the emphasis correct, you'll begin to speak with the same rhythm as native speakers. This inherent rhythm is something that will reduce your accent and make you more understandable to other English speakers.
Another cause of heavy accents when speaking English is the lack of native intonation. Intonation is the melody of the language, with pitch patterns appearing in sentences and phrases. The intonation used when speaking helps to put forward the meaning of the content of a sentence, and unusual intonation patterns can cause the hearer to misunderstand what you're saying to them.
For American English, it's the intonation that conveys the mood of the speaker. High-pitched speaking often denotes the speaker is surprised, while a low pitch can mean anger. No pitch, as in speaking in a monotone, can be seen as showing boredom with a topic.
By practicing intonation, your English will sound more natural to the hearer. Find a script or play and practice reading lines that have obvious moods using the proper intonation for the sentences. A common use of intonation in English is a rising and falling pattern. Speak with a rising tone to emphasize the most important word and then allow your tone to fall when you're finished speaking. For example, in the sentence "When is he coming?" your pitch should rise and fall on the word coming. For inquisitive sentences, you should use a rising intonation pattern to show you're waiting for an answer. For example, if asking "Is the pizza here?" you should raise your tone at the word here, and not lower it.
The best way to practice intonation is listen to how native speakers around you use tones in everyday speech and mimic it. However, once you begin to use the correct intonation in your speech, you'll speak with a much less noticeable accent.