Apartment in Portugal with internet connection?

If you want to stay in Lisbon, there is a private company which I have used to rent an apartment twice. Both times the apartments came with free Wifi internet access. This is them.

Depending on the amount of internet traffic needed you might just buy a mobile internet SIM. Vodafone offers just a SIM for around € 15 and I think 10 hours/300 MB is included (€ 50 with hardware), but there are similiar offers from other providers. You can always top the card up when the limit is reached.

Would be worth considering as an alternative because many cheaper places might not have internet, so you wouldn't limit yourself that much.

Many places also have public WiFi (city centers, parks, public offices, sights - usually signposted), Portugal is pretty far advanced compared to other countries.

I've been to Carvoeiro last year and highly recommend it. Smaller town, indeed touristy, but very bearable. Free WiFi in the city center, pretty fast, too (was lower season though).

travellers cheques accepted as part payment

Overlooked this. Is there a specific reason for this requirement? TCs are very uncommon in Europe these days and especially when used to actually pay for something rather than to simply exchange into cash.

You could cash the cheques yourself and then pay the landlord in cash. AFAIK getting a cheque credited might take a few weeks for the landlord and some may consider that too risky because of possible fraud, so you might have a hard time finding someone who is meeting this requirement.

many thanks to both of you. re: travs cheques - i was given them as a present 2 years ago and have never found a way to get rid of them so i thought this holiday would be a good opportunity but i'm prepared to have to cash them in or hang onto them until i find somewhere in europe that will accept them!

I doubt anyone anywhere would accept TCs as payment for something because you often get charged commission on changing them into cash so you'd have the problem of paying for the full amount or getting your change.