When I speak English, I can't tell the difference between cycle and psycho, I pronounce them the same.
And it's not only cycle vs. psycho; when words end in -le or -o, I always confusee them.
How to pronounce them correctly?
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The second syllable of cycle has the syllabic consonant (=acts like a vowel) l. Your tongue should be the front of the palate at the alveolar ridge, perhaps extending a bit to the teeth. You hold it there and create the syllabic center, even though it is traditionally classed as a consonant.
The second syllable of psycho is quite different, in that o it is a rounded vowel, and your tongue should not be touching your mouth anywhere.
Though I love tchrist's answer regarding the mechanics of pronouncing the words differently, I think that the fact is simply: there's an "L" in cycle. If you don't pronounce the "L", for whatever reason - cultural, lazy style, regional dialect - you will always stand the chance of being misunderstood, particularly by native speakers.
You mention "American English", so I suspect you are having a difficulty with the 'dark L'.
There are two (or more) L-sounds in (many varieties) of English: clear L and dark L.
Clear L is pronounced much like it is in French, Spanish, or German: /l/ in IPA. In American English it's usually found as initial L in words like leaf for example.
Dark L, /ɫ/ in IPA, is pronounced with the tip of the tongue touching the roof of the mouth behind the teeth, with the back of the tongue retracted back toward the throat. The sound is very much like /w/. In American English it's often found as final L in words like full or bottle.
You might be confusing /ɫ/ with /w/ in your speech. The way to avoid this is to ensure that in /ɫ/ the tip of your tongue touches the ridge behind your teeth.
Note: the distribution of the clear vs. dark L in various dialects of English is pretty complex so it's not surprising you may have a problem.
I can only speak for my own accent here; but for me, the -le ending and the -o ending are completely different.
To say -o as in psycho, my tongue stays low in my mouth, and the corners of my lips move inwards from the sides, making a slightly circular shape.
To say -le as in cycle, my lips don't move, but my tongue comes upwards and strikes the roof of my mouth. Most importantly, I continue voicing the sound until just after my tongue has struck the roof of my mouth - I don't let it tail off. That way, the L sound is clear for everyone to hear.
It will depend on the particular accent. There is an old joke that relies on it being possible to confuse the two, at least in some accents:
So two bits of black tarmac, and some green tarmac walk into a bar. The barman refuses to serve first bit of black tarmac: "Sorry, we don't serve surfacing materials here." The same happens to the second bit of black tarmac. So another customer asks what's going on when the barman serves the green tarmac, and the barman replies; "I'm not going to say 'no' to him, am I? He's a bleedin' cyclepath".