I'm working of a quite a large support organisation and we have a strong reason to believe that our outgoing emails are being delivered to spam folders.
Are there any services I can use to check if this is actually the case?
Next, check the Reverse DNS on the IPs which send email for your organisation. If they don't resolve to the domain sending mail then you'll want to check your domain's SPF records to ensure any authorised servers are listed.
Outside of having some dummy accounts set up at Yahoo/Hotmail/Gmail/etc for testing, you can use a 3rd party service like Return Path. I've not used them myself, but they offer a "Certification" service to help get you whitelisted.
I am using email on Acid: http://www.emailonacid.com/email-preview/deliverability.
I wouldn't trust its results in 100% (as spam filters constantly evolve) but it can give you a quite good idea what may be wrong with your emails.
MailChimp has a good tutorial here: http://kb.mailchimp.com/article/avoiding-the-spam-filters.