How do I produce a linearly chirp signal in spice?
To produce such a signal one way would be to use an arbitrary voltage source:
Syntax: Bnnn n001 n002 V = [expression]
This is a source with a function you define.

Here is an example of sweeping from 1 to 20Hz in LTSpice:



And the simulation:



I have used the time variable (defined in LTSpice as current sim time) to change the frequency (the (19 * time) part ) to keep things simple, but you can obviously use some other variable to change things (e.g. V or I of another circuit element)    

Here is the netlist in case you want to cut/paste:  

R1 N001 0 1k
B1 N001 0 V=sin(2 * pi * time * (1 + (19 * time)))
.tran 0 1000m 0
.backanno
.end