I believe you will be able to use a PIC to control DRAMs.

I've seen an 8051 chip receive and store faxes broadcast over
teletext data packets. The DRAM modules were several megabytes.

There are many clever ways of using DRAMs.
The most suitable one depends on your application.
I'd guess you wish to store a fairly big one or two-dimensional data
either audio or video.

If you are stuffing data in at a rate low enough for software control,
then you can control DRAM refresh in software easily.

The're simple enough in operation:
RAS refreshes the charges every capacitor on a particular row, and
CAS connects one particular capacitor to the column sense amplifier.

Thus if you miss out the RAS, you get a purely refresh cycle.
If you do multiple CAS pulses, you get multiple column access (page
mode).

The primary requirement of DRAMs is that all rows get accessed within
the refresh period.

Some applications make regular memory accesses: e.g.
oscilloscope data buffers, framestores, video memory etc.

The 6502-based BBC micro used DRAM, and the regular memory accesses of
the 6845 CRTC chip effectively refreshed the memory as it refreshed the
video image.

If you have something like a digital audio delay line, this can also do
refreshing as a side-effect.

I've designed circuit for using DRAMs with the 6502 (9 TTL chips,
including two 4464) and the 8085 (circular buffer for storing
pre-trigger data streaming out of EEG sensors).

Tell us more about your app.

Perhaps other readers might be able to come up with other useful
applications for the fast cache SRAMs and DRAM modules abundantly
available from obsolescent PCs.

Cheers, Keith.