I've often heard this kind of sentence where one substitutes the conjunction "like" for "as". Is it acceptable in written English? Is it considered wrong in spoken English?
As Bryan Garner puts it (in GMAU, 3e): "It is acceptable in casual English; it isn't yet in the category of unimpeachable English." In other words, don't write this way if you want to be considered educated. But, if you correct someone who speaks this way, you'll probably be considered a "grammar Nazi".
My observation would be that the usage you mention is common and acceptable in spoken/informal usage. But in formal or "careful" usage, it might sound a little out of place to some ears.
One slight point to bear in mind is that "like" isn't always analogous to "as": it is also commonly used to mean "as though"/"as if" (the latter are essentially synonymous, but "as though" is a little more positive). In informal usage, there's usually little ambiguity. But in formal usage, if you avoid "like" and instead opt for "as"/"as though"/"as if", this gives you the opportunity for a little more precision.
The other thing to bear in mind is that (unlike, say, "whom", which is overtly formal) "as"/"as though"/"as if" probably won't sound overly formal to most people if you use them in informal speech. So if you're in doubt, you could just use "as"/"as though"/"as if" and avoid "like" with this usage.