Maybe it's "“offline search” (and gadgets) program", but Linux version doesn't include gadgets, so far...

Here comes screenshots by Linux user in the comments:

Google Desktop adds its own global category to the Gnome Menu (grrr!)
http://www.trammer.pl/ludwik/screens/google-desktop/menu.png

Icon in tray and its menu
http://www.trammer.pl/ludwik/screens/google-desktop/tray.png

Quick search box
http://www.trammer.pl/ludwik/screens/google-desktop/quick-search.png

Homepage (old Google design)
http://www.trammer.pl/ludwik/screens/google-desktop/search.png

Preferences
http://www.trammer.pl/ludwik/screens/google-desktop/preferences.png

Yeah, it's a bit disappointing compared to where Google desktop is for Windows and Mac users. There's not really much to show. Think of Google desktop 1.x for windows, that's what this looks like.

Right now, I am happy with Beagle on Linux. For me, one of the main perks of Linux is that it is open source – programs develop with demand and innovation is that much more likely. Not only does Google Desktop for Linux not have open source code, it could harm the open-source desktop search engines. As helpful as it may be, this will now be added to my (short) list of Google products that don't get used by me too often.

this could become interesting once it will advance to a level of where GD 5 is today

I agree with Eytan. I unistalled Google Desktop for Linux and installed Beagle to use it with deskbar (it's gorgeous!). Much nicer than Google Desktop.

I installed it last night and was a little disappointed to find that is was more like GD1.x for windows. I decided to let it start indexing and went about my business and firefox crashed about 7 times....I uninstalled it and restarted all my beagle processes and everything is happy again. This might be one of the only Google "Beta" softwares, that actually acted like a beta software for me. I may try again in a few versions. BRING ON GOOGLE TALK (with talk capability) for linux :)

I'm sure better versions are going to come soon for Mac and Linux both. First step is the hardest :)

I can't wait till they finally create a Mac client for GTalk. Skype seems to be the best Mac client with the least amount of headaches when trying to use the voice chat feature. Plus it also has video capabilities. Too bad most of my contacts use different IM clients.

Regardless of the it's apparent "version 1" behavior (based on the experiences of those who have tried it), It's still great to know that Google is catering to the Linux market, no matter how "small" this is. I didn’t like version 1 of GDS for windows too, (and to be honest, MSN Desktop Search was indeed better than GDS during the early versions), but I guess everyone knows the rest of the story; how Google stepped on the pedal and brought significant improvements to GDS in amazing speed. So I guess this fact should be taken into perspective. But really, the mere fact that Google has taken the time and resources to deliver a GDS version for the linux platform is already a laudable act as its. (Of course, they’re doing this with an intended benefit for them). And knowing how Linux is well loved in the Google campus, this should only be the beginning.

Google says it has "almost" all the features from version 1...

the main feature that seems to be missing is the ability to launch programs and websites from the ctrl ctrl box (which was not a feature in the original gds). I may be able to add the program launch feature by indexing /usr/bin and a couple other directories like /opt though, Maye a later version will index the DE menu by default – that would be a little more de specific and would probably increase the requirements, but they already do that requiring that the desktop comply with freedesktop standards for the tray icon, and I believe those same standards govern the means of adding, removing and accessing entries in the equivalent of the start menu.
I think the reason why they haven't focused on gadgets for the mac/linux versions is that both OS's already have a system for that built in – mac has the dashboard and linux has the panel applets. Windows just recently got the sidebar with Vista, and that seems to be partly a response to google's desktop sidebar. This doesn't necessarily mean that we won't get them, but that they don't take as high a priority.
I'm still undecided on gd vs. beagle. I tried beagle a while ago but it didn't seem to have a search as you type, which is one of my favorite features of gdesktop. Maybe I'll reinstall Beagle to do a better comparison, and keep checking back on both pieces of software for improvements. Regardless, I'm thrilled to see google port another piece of their software to linux (and this one seems to be a real port like google earth rather than a wine-packaged version like picasa), especially when you consider the market share of linux compared to the windows features they could have developed in that time. But then again, this aides in business-wide deployment of gds (I wonder how long until we see gds enterprise for linux and mac), and while they're a minority, early adopters like linux users are a key to the success of new google services.

In checking out the Beagle page, I found an interesting piece of irony; Beagle is a participant in the Google-sponsored Summer of Code. How's that for helping the competition

I've just installed an update but I don't know what it brings.