Kini/kito/kin - whither/hither/thither: For all its apparent simplicity, Iban has these directional location words whose equivalents English more or less lost long ago. "Kini di" (whither you, i.e. where are you headed) is a common greeting, and the proper response is simply "kin", accompanied by a vague gesture.
Enggaika - lest: Similarly, Iban has a nice simple way to say lest, combining enggai - literally "don't want" with ka, an ending indicating causation or direction of action roughly equivalent to the Malay ending -kan. Like whither or hither, lest is a bona fide English word, but is used extremely rarely (at least in America), whereas enggaika is a reasonably common Iban word.
Ngambika/awakka - These words are the positive counterparts of enggaika, expressing the idea of "in order that" or "leading to." They are roughly equivalent to "supaya" in Malay, and have the same -ka ending as enggaika.
Udah - already: Very common word, equivalent to "sudah" in Malay, meaning not just already but really simply indicating a completed state. Udah can be stated more or less as a complete sentence, as in "when you will clean your room?" "udah."
Kelia - long, long ago: Iban culture has an epic poetry tradition as well as its share of unique religious practices and beliefs. Kelia is a reasonably common Iban word that I think communicates a flavor of this. I've heard that some languages have differents sets of words and tenses for events in the "mythical" past as opposed to the simple, experienced past. I think that kelia is something like that for Iban, since "dulu" is good enough to indicate "long ago," but kelia exists to indicate a time much longer ago, in the distant, prehistoric, or (I conjecture) mythical past. I heard it used in Christian settings for stories about ancient prophets.
Nyiur - coconut: Delicious and omnipresent on Borneo. I think the word is interesting because this word indicates Iban's common heritage with other Austronesian languages, in particular Hawaiian, in which "niu" (pronounced very similarly) is the word for coconut. It's easy to think of Malaysia as a part of Southeast Asia, and therefore culturally similar to places like Thailand and as a part of "Greater China." I think it's important to remember, however, that Borneo is more of a Pacific Island than it is an outgrowth of Asia/China.
Rumput - grass: This is one of several words for grass, and it's the same (and more commonly used) in Malay. It's actually a word that doesn't sound or seem similar to most other Iban and Malay words, at least to my ear. I read once on the website of a believer in linguistic monogenesis that this word is essentially the same as the word for grass in Egyptian. I don't have a citation for this but I found the idea intriguing - that Iban and Malay are distant descendants of a common human tongue, distorted beyond recognition but having retained in common with one of the most ancient languages a single word. Most people don't believe anything like this, chalking it up to coincidence, but I like the idea and I don't think it impossible.
Seremang- point (insultingly). There were several major Iban rules of etiquette that I encountered. One was to never step in front of someone who was sitting, one was to never refuse a host's food, and one was to never point with a finger like Americans do. There is even this word for pointing insultingly.
Sebilik- (nuclear) family. In Malay, this would literally mean "one room." Ibans have traditionally lived comunally in "longhouses," in which each nuclear family lives together in one room along a great hall. So naturally, sebilik is family.
Bilun- airplane. From the English "balloon."
Gostan- go backwards. From the English "go astern."
Heli- helicopter. From the English, shortened, but pronounced in an almost un-understandable way.
In addition to diphthongs, there were plenty of times that the vowel simply changed: "putih" became "puteh," "emahk" became "mac," "untuk" becam "untok." Sometimes this vowel change is reflected in orthography. I remember a hospital that I walked by sometimes in Singapore, where the Malay translation of "hospital" on the sign was "rumah untok orang sakit," funny both because of the spelling "untok" and because of the many syllables needed for "hospital." This orthography shows up in Malay/Iban cognates, even in dictionaries. Richards' Iban dictionary has "itong" as the word for what in Malay would be "hitung," for example.