This is kind of a tough question, as there are various different aspects of Canon DPP and Lightroom/ACR that offer both pros and cons for each tool. Both DPP and LR/ACR will produce better results than Picasa, which while it does support RAW conversion, is not a top of the line tool explicitly designed for converting RAW files (it's more of a photo management tool than anything.)
Canon Digital Photo Professional
When it comes to Canon DPP, you'll probably find that it offers a lit better control over the shadows, and in some cases better noise reduction. DPP is straight from Canon, the designers of the CR2 raw format, and they maximize the dynamic range potential and electronic noise removal potential in their own tool. If you must extract the most dynamic range possible from your files, or tend to have problems with fixed pattern noise, DPP will generally do a slightly better job than ACR. DPP has also offered color curve editing for a while, a feature that has only recently made its way into Lightroom, albeit in a minimalistic way. In the past, the difference was significant enough that many Canon photographers would use DPP despite hating its UI, however with Lightroom 4.x and ACR 7.x, things are improving for Canon in those tools.
There are a few drawbacks to using Canon DPP. First and foremost, its not the greatest tool from a workflow or UI layout standpoint. It can at times be quite frustrating to work with. Another issue with DPP is its demosaicing algorithm. While it tends to handle dynamic range and noise well, it is not as good as LR/ACR at avoiding color moire in some cases, and for fine detail such as hair, dimples, pine needles, etc. it tends to produce semi-aliased edges. If you only have Photoshop+ACR 6 or Lightroom 3.x or older, then DPP does have quite a bit to offer...however it definitely comes with trade-offs.
Photoshop w/ Adobe Camera RAW and Lightroom
While Canon DPP has a few key things to offer, such as better DR when editing CR2 files, it lacks a lot compared to Photoshop+ACR or Lightroom. Both Photoshop and Lightroom have better UIs, and offer far more extensive tool sets with which to process your photos. Lightroom offers a very significant advantage in terms of workflow and library management. With the latest Lightroom 4.x releases, Lightroom brings several new features that are not present in DPP, such as print soft-proofing. Lightroom's newest raw processing engine, used in ACR 7.x and Lightroom 4.x, greatly enhances control over the full tonal range present in a RAW file. Particularly in the highlights, but also in the shadows. This closed the gap between LR/ACR and DPP, however it has not yet given Adobe's products an edge when it comes to extracting the most out of the available dynamic range. An area where LR/ACR excel is their demosaicing algorithm. It is a very refined algorithm that is capable of eliminating most color moire, and it is capable of producing very clean, sharp edges especially along fine detail like hair.
From a cons standpoint, LR/ACR are not as capable at removing noise, particularly electronic noise contributions, from CR2 files. Canon DPP still does a better job in this area. I've also had a hard time determining whether the latest LR/ACR does a better or worse job removing noise than Lightroom 3.x...in some cases it seems slightly better, in other cases LR4.x and ACR 7.x seem to blur a lot of useful detail away when removing noise that LR 3.x did not. When it comes to dynamic range, and extracting detail from the deep shadows, LR does not seem to do the CR2 format justice. Despite considerable improvements in Adobe's latest raw processing engine, Canon RAW files still seem to suffer quite a bit on the low end of the dynamic range. They fare quite well in the highlights, and LR 4.1 is able to extract an astonishing amount of detail from what one would normally think were completely blown white highlights. Shadow pulling, however, seem to greatly enhance noise, and if you have any fixed pattern noise (FPN), it readily comes to the surface when working shadows. FPN is usually only a problem at the lowest ISO settings...beyond ISO 400 noise is mostly photon shot noise and very random. At ISO 100 and 200 can often exhibit FPN right into the lower mid-tones (something that doesn't occur that often in DPP.)
Personally, I use Lightroom for my post-processing work. Despite its drawbacks in terms of handling Canon CR2 files and extracting the most out of dynamic range, and its sometimes lackluster noise removal, it has so much more to offer that I'm willing to accept those two drawbacks. Since Lightroom handles highlight dr so well for CR2 files, you can often mitigate the problems with shadow dr by exposing to the right (adjust exposure so it bunches up more in the highlights than the shadows, without oversaturating as much as possible). You can easily pull down highlights and have more dynamic range in the shadows to work with. You'll never realize the same amount of DR as you could extract from a camera using a Sony Exmor sensor (like the Nikon D800), but you can improve your options. If dynamic range or FPN removal is of paramount importance, you might want to look into using DPP, at least in those cases where you need to perform a lot of shadow lifting. You might take a hit to fine detail edge quality (and you might find that fine detail in general becomes a little more blocky), but you should see better results in the areas where LR is weakest.