A possible quick and easy answer to your original question that I found while researching my reply is to download the narcoleptic library from here http://code.google.com/p/narcoleptic/
and just sleep the arduino for x amount of time. The more complex answer if the sleep time needs to be accurate is to use the SQW output from your DS1307 to wake the arduino every second (or less) and you count the number of wakeup's and do something as required (eg every minute (60 wakeup's) read the temperature)
The principle is the same for both the shield you bough and the Chronodot, the difference is in the RTC chip being used. The DS1307 has a SQW output pin that is configurable for 1Hz, 4kHz, 8kHz or 32kHz output. The DS3221 SQW pin is configurable for either 1Hz or ALARM output. I would start with the shield you have and sleep the arduino for 1 second at a time, waking it with the SQW interrupt set to 1Hz. Once you get this working okay then substituting the DS3221 should only entail minimal hardware changes and a bit more programming to set alarm times and configure the SQW pin as an alarm interrupt instead of square wave output.
I cannot find a DS3221 library that has the alarm setting commands you would require but maybe someone else knows of one.
I would use this library https://github.com/davidhbrown/RealTimeClockDS1307
for the DS1307. I have used it myself and it works really well and also has the SQW configuration commands you will need.
I cannot find a schematic of the shield you have and cannot determine if/where the SQW pin is broken out on the shield pins. You will need to read up on causing the arduino to sleep and wakeup using an interrupt. I recall Nick Gammon doing an article on power saving using processor sleep, he and others can maybe give you more info on this.
*I'm quite new to arduino though not micro controllers (PIC) and have not ventured to wake on interrupts yet.