Development of a fully-featured CAT tool solely for Linux is probably not a commercially viable venture; even Mac users have waited in vain for years for a dedicated CAT tool for their platform of choice. Nor have any of the major commercial CAT tool vendors ported their products to Linux. Linux users determined to use one of the market leaders' products (notably SDL Trados or MemoQ) are known to have resorted to running these Windows products in an emulator.
A plethora of CAT tools exist however that are able to run in a virtual machine (typically Java), and therefore on a variety of operating systems; some of the less comprehensive Windows CAT tools are also able to run on WINE.
Notable examples of Java-based CAT tools are:
OmegaT (free and open source), popular (owing to the lack of a price tag) with professional translators who are new to CAT, and with a strong following among volunteer translators in the open-source community; OmegaT has also been adopted, and forked for separate development, by the European Commission's Directorate-General for Translation (DGT)
Cafetran, a small commercial vendor that has nevertheless been able to build up a loyal and enthusiastic following
Wordfast Pro, an offering by what was at one time one of the major commercial vendors, but which has failed to capture the market share that its sister, Wordfast Classic, enjoyed in the latter's heady days two decades ago
Swordfish and its sister products, a commercial product from Maxprograms; very much the "tech's" CAT tool
The Heartsome suite, by the same developer as Swordfish: Heartsome has now been released to the community and is available free of charge
Of Windows CAT tools that can be made to run on WINE, Wordfast Classic is doubtless by far the most popular.
As with office suites, compatibility is an issue in the suitability of a particular CAT tool. The relevant file formats are however now largely XML-based, and fitting a given CAT tool into the productivity chain (i.e. meeting customer requirements) generally much easier than was once the case.