There's nothing inherently wrong with using AutoNumber, but there are a few reasons not to do it. Still, rolling your own solution isn't the best idea, as dacracot mentioned. Let me explain.
The first reason not to use AutoNumber on each table is you may end up merging records from multiple tables. Say you have a Sales Order table and some other kind of order table, and you decide to pull out some common data and use multiple table inheritance. It's nice to have primary keys that are globally unique. This is similar to what bobwienholt said about merging databases, but it can happen within a database.
Second, other databases don't use this paradigm, and other paradigms such as Oracle's sequences are way better. Fortunately, it's possible to mimic Oracle sequences using SQL Server. One way to do this is to create a single AutoNumber table for your entire database, called MainSequence, or whatever. No other table in the database will use autonumber, but anyone that needs a primary key generated automatically will use MainSequence to get it. This way, you get all of the built in performance, locking, thread-safety, etc. that dacracot was talking about without having to build it yourself.
Another option is using GUIDs for primary keys, but I don't recommend that because even if you are sure a human (even a developer) is never going to read them, someone probably will, and it's hard. And more importantly, things implicitly cast to ints very easily in T-SQL but can have a lot of trouble implicitly casting to a GUID. Basically, they are inconvenient.
In building a new system, I'd recommend using a dedicated table for primary key generation (just like Oracle sequences). For an existing database, I wouldn't go out of my way to change it.