Those are the correct SPI connections.
However, some of the newer shields get SCK, MISO & MOSI on the 6 pin socket rather than pins 13, 12, 11. If you have one of those, you'll need to make the connections from the 6 pin header over to Teensy's SPI pins.
The W5100 chip also requires a reset pulse. Sometimes it will work without a reset, other times it will play dead until you reset. Normal Arduino connects the AVR reset to the shield, so the W5100 gets a reset when you upload. The simplest approach is to just wire the reset pin to any of Teensy's digital pins and add a little code to drive the pin low for a few milliseconds, then high (which allows the W5100 to run).
The Wiznet chip uses quite a lot of power. I believe most of the Arduino Ethernet shields have a regulator on-board to make the 3.3V power, since Arduino boards can't output enough power on their 3.3V pin. If you haven't wired 5V to the 5V and VIN pins on the shield, it's quite possible your W5100 is sitting there without any power. This is pretty simple to check with a voltmeter on any of the capacitors or parts near the chip.
Most of those shields have a SD card socket. Do not put a card in the socket, at least not until you get the ethernet part working. I believe almost all the ethernet shields for regular Arduino lack pullups on the CS signals, so the SD card "hearing" the ethernet communication is a problem if pin 4 isn't configured, or isn't even connected to Teensy. Simplest to just make sure you don't have a card in the socket.
I can tell you I have personally verified both V2 and V3 Arduino Ethernet shields. It will work when you get the connections correct! Sparkfun makes a shield adaptor, and it is known to have compatibility problems with the ethernet shields. Somewhere on the forum there's an old thread where I documented the issues. (hopefully Sparkfun has updated their adaptor by now?) With the changes, it definitely did work with the ethernet shields.