I'm thinking it makes sense to run your Tumblr from mycompany.com/blog so the traffic is associated with your domain, but are there any reasons to leave it at mycompany.tumblr.com?
The big advantage is future flexibility. If in 5 years time, Tumblr closes down you lose everything - all the SEO value of your content, your customers' bookmarks, etc. You can migrate the content, but you'll be starting from scratch with marketing.
If you put it on your own domain, and Tumblr closes down, you can switch to Wordpress.com, migrate the content, format the links in the same way, and nobody need even know that you have moved.
The only disadvantage of having it at mycompany.tumblr.com is the extra maintenance involved. You now have to maintain the domain name and the DNS settings, and keep an eye on the security of your domain name, which Tumblr does for you when it's on their domain.
However, this isn't a big deal - you can register the domain for 10 years at a time, and you should be doing the DNS stuff anyway if you already have a website at mycompany.com.
The advice I usually give is - at some point you will have to do this, and the sooner you do it, the smaller the headache involved.
Useless answer but a must. It depends.:)
If your company or web site is strong, I do not see much reason to use mycompany.tumblr.com.
First, there is a distance between your web site and tublr blog for your visitors. Even though it is just one click away between them, but it is still there. If your visitors go to your tublr blog, will they come back? not sure.
Second, if you company is strong while you put your official blog on their platforms, it seems you are not committed to your community or customer communication to me.
The last one I can think of though is not so important for some web sites,but if you have blog like mycompany.com/blog, and update it frequently, search engines will love that, this is especially true for those web sites with no fresh content.
On the other hand, the upsides of using mycompany.tumblr.com is you will get lots of back links to your web site, and some time you may get more visitors from tublr users, but I am not sure. Also you have a whole lot of options to customize your blog very easily like using different themes.
If your company or web site is small, maybe it make sense to use tublr as your blog, if you company is big, use your own domain and migrate tublr content to your domain blog.
By the way, our company is using both.