After taking a peek in the Wine
AppDB I found that Deus Ex was listed as working well. Given that I'd heard so much about and never had a chance to play it I went out and purchased a copy (GOTY edition).
After installing, it evidently wants you to have the CD in the drive. After a quick google for NoCD cracks, I realized that one of the cracks was just an .ini file with one main difference.
I changed this: CdPath=Z:\media\DeusEx_GOTY_1
To this: CdPath=..\
Eh voila, no more CD check.
Ah, for the days of yesteryear when copy protection was just an afterthought by the dev and easily worked around. It certainly helps for the filthy Linux using degenerates among us anyway.