Diode logic computer
It's certainly possible to make a computer without transistors, using diode logic for most functions. All computers prior to 1953 avoided transistors, and some of these heavily used diode logic.

But eventually you need some form of amplification and inversion.

Inversion you can easily achieve using transformers (at least, if you are passing discrete pulses rather than continuous logic levels through the logic. This was common in the 1940s and 50s) - simply swap the secondary winding connections.

Amplification : assuming you have ruled out valves (vacuum tubes) as well as transistors, you are limited in your options. Relays are an obvious choice, for clock rates up to a few Hz. Above that, there are tricks you can play on transformers to amplify current changes by using smaller currents in other windings to bring their cores in and out of saturation. I've never heard of anyone exploiting this form of "magnetic amplifier" for computing, so it may not be possible. 

On the other hand, the Elliot 803 was a transistor computer, but it implemented its logic functions using magnetic cores, with just one transistor per gate to provide gain.